Question

解答数学题高中数学必修一 已知函数fx=x+1/x 有定义证明fx在（0，1）上是减函数

Answer 1

设0<x1<x2<1，
则f(x1)-f(x2)
=x1+1/x1 -(x2+1/x2)
=x1-x2 +(x2-x1)/(x1·x2)
=(x1-x2)[1- 1/(x1·x2)]
=(x1-x2)(x1·x2 -1)/(x1·x2)
因为0<x1<x2<1，所以 (x1-x2)<0，(x1·x2 -1)<0
所以 (x1-x2)(x1·x2 -1)/(x1·x2)>0，
即 f(x1)>f(x2)
从而 f(x)在(0,1)上是减函数。

Answer 2

证明：在其定义域（0.1）中任取x1,x2且x1<x2. 则f(x1)-f(x2) =x1+1/x1 -(x2+1/x2) =x1-x2 +(x2-x1)/(x1·x2) =(x1-x2)[1- 1/(x1·x2)] =(x1-x2)(x1·x2 -1)/(x1·x2) 因为0<x1<x2<1，所以 (x1-x2)<0， (x1·x2 -1)<0 所以(x1-x2)(x1·x2 -1)/(x1·x2)>0， 即f(x1)>f(x2) 从而f(x)在(0,1)上是减函数。呵呵，一样们帮到你